Superfluid nanomechanical resonator for quantum nanofluidics

We have developed a nanomechanical resonator, for which the motional degree of freedom is a superfluid ${}^4\mathrm{He}$ oscillating flow confined to precisely defined nanofluidic channels. It is composed of an in-cavity capacitor measuring the dielectric constant, which is coupled to a superfluid Helmholtz resonance within nanoscale channels, and it enables sensitive detection of nanofluidic quantum flow. We present a model to interpret the dynamics of our superfluid nanomechanical resonator, and we show how it can be used for probing confined geometry effects on thermodynamic functions. We report isobaric measurements of the superfluid fraction in liquid ${}^4\mathrm{He}$ at various pressures, and the onset of quantum turbulence in restricted geometry.

Figure 1

The superfluid nanomechanical resonator is defined by an oscillating mass of superfluid ${}^4\mathrm{He}$ confined in the channels of a nanoscale structure. (a)–(d) The nanofabrication process for our devices (see Appendix pp1 for details). (e) Photograph of the completed device with an effective cavity height $h_{\mathrm{eff}}\simeq 900$ nm and a channel height $h_{\mathrm{cha}}\simeq 550$ nm. The light blue, yellow, and purple colors are the result of optical interference.

Figure 2

Measurements of the dielectric constant ${\epsilon }_r$ of 11 nL of liquid ${}^4\mathrm{He}$ confined between the electrodes of the nanofluidic cavity, from 4 to 1.65 K at saturated vapor pressure (blue), 5 bar (orange), and 10 bar (green). Circles are bulk values at saturated vapor pressure from Ref. [28]. The $\lambda$ transition occurs at the kink in ${\epsilon }_r$.

Figure 3

Simplified schematic of the superfluid nanomechanical resonator. (a) The electrostatic force, $F_{\mathrm{stat}}$, deforms the glass cavity and generates a pressure gradient across the channel. (b) A schematic of an equivalent mass-spring system—the plate moves by $x$ and the mass of superfluid, $m_{\mathrm{He}}$, responds by moving a distance $y$.

Figure 4

Temperature dependence of the superfluid nanomechanical resonator (a) frequency and (b) quality factor, taken at a drive voltage $V_d=5$ V, for various pressures: 2 (blue triangles), 5 (orange squares), 10 (green diamonds), 15 (red circles), 20 (black diamonds), and 25 (purple diamonds) bar. The inset of (a) is the superfluid fraction, ${\rho }_s/\rho$, extracted from the resonance frequency using Eq. (1), with $T_{\lambda }$ taken from Maynard sound measurements [36], and the fit function (black line) given by Eq. (3). The inset of (b) is a frequency sweep across the resonance, which shows the real part (black circles) and imaginary part (blue circles) of the relative dielectric constant measured at 2 bar and 1.62 K and the corresponding fit functions obtained from Eq. (2).

Figure 5

Temperature dependence of the superfluid nanomechanical resonator (a) frequency and (b) quality factor, taken at constant pressure $P=5$ bar for various drive amplitudes: 2, 5, 7, and 10 V. The inset of (b) shows the quality factor measured at $T=1.70$ K (black arrow) as a function of the drive voltage. The inset of (a) is a frequency sweep of the in-phase (black circles) and the out-of-phase (blue circles) dielectric response at $V_d=10$ V with their fit functions obtained from Eq. (2).

Figure 6

Cross section of the channels having an electrode passing through (a) and without an electrode (b). The region of the nanofluidic device confining the liquid ${}^4\mathrm{He}$: a cavity and four channels (c).

Figure 7

Capacitance measurement (blue circles) of the nanofluidic capacitor under the application of a varying dc voltage (0–20 V), which bends the cavity glass plates and increases the capacitance $C$. These data are fit (gray line) to Eq. (D7) in order to extract the bending stiffness of the glass plates.

Figure 8

FEM simulation of the first mode of liquid ${}^4\mathrm{He}$ confined in the nanofluidic device. The color bar represents the acoustic pressure on resonance $({\omega }_0/2\pi \sim 7$ kHz) in arbitrary units. The mode shape is analogous to the Helmholtz resonance with four masses in the channels connected to an effective spring in the cavity. In this simulation, we increased the thickness of the structure by a factor of 1000 so the mode shape is easier to see.

Figure 9

Temperature dependence of the superfluid fraction, ${\rho }_s/\rho$, extracted from the resonance frequency using Eq. (1) with $T_{\lambda }$ taken from Maynard sound measurements [36]. Data taken at a drive voltage $V_d=5$ V, for various pressures: 2 (blue triangles), 5 (orange squares), 10 (green diamonds), 15 (red circles), 20 (black diamonds), and 25 (purple diamonds) bar. The black line is obtained by fitting the data measured at 2 bar with Eq. (F1).

Figure 10

Displacement of the superfluid mass $y$ induced by the electrostatic drive (red line) and the electrostrictive drive (blue line) as a function of the stiffness of the glass plates. In our geometry, the superfluid nanoresonator is mainly electrostatically driven.

Figure 11

Magnitude of the four forces acting on the superfluid nanomechanical resonator as a function of the gap $h_{\mathrm{eff}}$ between the electrodes for $V_d=1$ V. There is an electrostatic force $F_{\mathrm{stat}}$ (red line), an electrostrictive force $F_{\mathrm{strict}}$ (blue line), an electromagnetic Casimir force $F_{\mathrm{Cas}}^{\text{EM}}$ (green line), and a critical Casimir force $F_{\mathrm{Cas}}^{\text{crit}}$ (black line).